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Mirrors > Home > ILE Home > Th. List > uzsinds | Unicode version |
Description: Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.) |
Ref | Expression |
---|---|
uzsinds.1 | |
uzsinds.2 | |
uzsinds.3 |
Ref | Expression |
---|---|
uzsinds |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzsinds.2 | . 2 | |
2 | oveq2 5782 | . . . 4 | |
3 | 2 | raleqdv 2632 | . . 3 |
4 | oveq2 5782 | . . . 4 | |
5 | 4 | raleqdv 2632 | . . 3 |
6 | oveq2 5782 | . . . 4 | |
7 | 6 | raleqdv 2632 | . . 3 |
8 | oveq2 5782 | . . . 4 | |
9 | 8 | raleqdv 2632 | . . 3 |
10 | ral0 3464 | . . . . . . 7 | |
11 | zre 9065 | . . . . . . . . . 10 | |
12 | 11 | ltm1d 8697 | . . . . . . . . 9 |
13 | peano2zm 9099 | . . . . . . . . . 10 | |
14 | fzn 9829 | . . . . . . . . . 10 | |
15 | 13, 14 | mpdan 417 | . . . . . . . . 9 |
16 | 12, 15 | mpbid 146 | . . . . . . . 8 |
17 | 16 | raleqdv 2632 | . . . . . . 7 |
18 | 10, 17 | mpbiri 167 | . . . . . 6 |
19 | uzid 9347 | . . . . . . 7 | |
20 | uzsinds.3 | . . . . . . . 8 | |
21 | 20 | rgen 2485 | . . . . . . 7 |
22 | nfv 1508 | . . . . . . . . 9 | |
23 | nfsbc1v 2927 | . . . . . . . . 9 | |
24 | 22, 23 | nfim 1551 | . . . . . . . 8 |
25 | oveq1 5781 | . . . . . . . . . . 11 | |
26 | 25 | oveq2d 5790 | . . . . . . . . . 10 |
27 | 26 | raleqdv 2632 | . . . . . . . . 9 |
28 | sbceq1a 2918 | . . . . . . . . 9 | |
29 | 27, 28 | imbi12d 233 | . . . . . . . 8 |
30 | 24, 29 | rspc 2783 | . . . . . . 7 |
31 | 19, 21, 30 | mpisyl 1422 | . . . . . 6 |
32 | 18, 31 | mpd 13 | . . . . 5 |
33 | ralsns 3562 | . . . . 5 | |
34 | 32, 33 | mpbird 166 | . . . 4 |
35 | fzsn 9853 | . . . . 5 | |
36 | 35 | raleqdv 2632 | . . . 4 |
37 | 34, 36 | mpbird 166 | . . 3 |
38 | simpr 109 | . . . . . 6 | |
39 | uzsinds.1 | . . . . . . . . . 10 | |
40 | 39 | cbvralv 2654 | . . . . . . . . 9 |
41 | 38, 40 | sylib 121 | . . . . . . . 8 |
42 | eluzelz 9342 | . . . . . . . . . . . . . 14 | |
43 | 42 | adantr 274 | . . . . . . . . . . . . 13 |
44 | 43 | zcnd 9181 | . . . . . . . . . . . 12 |
45 | 1cnd 7789 | . . . . . . . . . . . 12 | |
46 | 44, 45 | pncand 8081 | . . . . . . . . . . 11 |
47 | 46 | oveq2d 5790 | . . . . . . . . . 10 |
48 | 47 | raleqdv 2632 | . . . . . . . . 9 |
49 | peano2uz 9385 | . . . . . . . . . . 11 | |
50 | 49 | adantr 274 | . . . . . . . . . 10 |
51 | nfv 1508 | . . . . . . . . . . . 12 | |
52 | nfsbc1v 2927 | . . . . . . . . . . . 12 | |
53 | 51, 52 | nfim 1551 | . . . . . . . . . . 11 |
54 | oveq1 5781 | . . . . . . . . . . . . . 14 | |
55 | 54 | oveq2d 5790 | . . . . . . . . . . . . 13 |
56 | 55 | raleqdv 2632 | . . . . . . . . . . . 12 |
57 | sbceq1a 2918 | . . . . . . . . . . . 12 | |
58 | 56, 57 | imbi12d 233 | . . . . . . . . . . 11 |
59 | 53, 58 | rspc 2783 | . . . . . . . . . 10 |
60 | 50, 21, 59 | mpisyl 1422 | . . . . . . . . 9 |
61 | 48, 60 | sylbird 169 | . . . . . . . 8 |
62 | 41, 61 | mpd 13 | . . . . . . 7 |
63 | 42 | peano2zd 9183 | . . . . . . . . 9 |
64 | 63 | adantr 274 | . . . . . . . 8 |
65 | ralsns 3562 | . . . . . . . 8 | |
66 | 64, 65 | syl 14 | . . . . . . 7 |
67 | 62, 66 | mpbird 166 | . . . . . 6 |
68 | ralun 3258 | . . . . . 6 | |
69 | 38, 67, 68 | syl2anc 408 | . . . . 5 |
70 | fzsuc 9856 | . . . . . . 7 | |
71 | 70 | raleqdv 2632 | . . . . . 6 |
72 | 71 | adantr 274 | . . . . 5 |
73 | 69, 72 | mpbird 166 | . . . 4 |
74 | 73 | ex 114 | . . 3 |
75 | 3, 5, 7, 9, 37, 74 | uzind4 9390 | . 2 |
76 | eluzfz2 9819 | . 2 | |
77 | 1, 75, 76 | rspcdva 2794 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wsbc 2909 cun 3069 c0 3363 csn 3527 class class class wbr 3929 cfv 5123 (class class class)co 5774 c1 7628 caddc 7630 clt 7807 cmin 7940 cz 9061 cuz 9333 cfz 9797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-inn 8728 df-n0 8985 df-z 9062 df-uz 9334 df-fz 9798 |
This theorem is referenced by: nnsinds 10223 nn0sinds 10224 |
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